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Weakly nonlinear shoaling of tsunami: the phenomenon of self-induced transparency

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HY2W01 - Modulation theory and dispersive shock waves

The evolution of tsunamis is reasonably well described by weakly nonlinear models which treat them as long low-frequency surface waves interacting with the topography. While in the deep ocean tsunamis are in general small-amplitude perturbations (of the order of 10 cm or less), their height may increase considerably through shoaling and refractive focusing. One of the key factors determining the outcome of a tsunami for particular shore is the share of the energy which passes through and reflected, called the transmission and reflection coefficients. Often the observed tsunami reflections might be substantial, up 90% of the incoming energy. Hence, it is important to understand and properly describe tsunami redlection and transmission. In one of the most common scenarios   tsunamis disintegrate into `solibores’ (trains of cnoidal waves/solitons), or, in other words, form a dispersive shock. This process has important implications for the threat of tsunami to coastal communities. The shoaling solibores grow faster than linear waves, as a result they become significantly steeper and attain higher-amplitudes than the tsunami source. They focus high velocity and accelerations in small packets that create a more effective forcing for nearshore currents and sediment transport. One of the main key open questions concerned with tsunami shoaling is how nonlinearity affects the incident wave reflection and transmission. Here, by investigating numerically within the framework of the Boussinesq equations the disintegration of a shoaling pulse into solibore, we show that the emergence of a well developed solibore considerably reduces the incident wave reflection by the bottom slope, or, alternatively, increases the total mass and momentum transmission passing through the slope and onto the shoreline. The effect is explained by the disintegration of an initially long pulse of a characteristic scale with a substantial reflection from the slope into a solibore made of robust nonlinear waves of much shorter scales. These shorter scales experience very little reflection. Thus, a finite amplitude largely pulse passes through, while a small amplitude one is reflected. In nonlinear optics and plasmas a superficially similar phenomenon when a small amplitude wave cannot get through, while the finite amplitude one can, is called `self-induced transparency’, we apply this term to our context. We conclude by stating that the described effect of self-induced transparency caused by disintegration of tsunami into solibores considerably increases its damaging potential.   Co-Authors and Affiliations: Joseph Jefferson and Alex Sheremet

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